Question

Degree of the differential equation $y = x\frac{dy}{dx} + a\sqrt{1 + (\frac{dy}{dx})^2}$ is

• A. 4
• B. 3
• C. 1
• D. 2

Hint:

Radical sign over a derivative? Square it! The degree is only defined for differential equations in polynomial form.

Answer 1

The Correct Option is D

Step 1: Concept
The degree is the power of the highest-order derivative after the equation is cleared of radicals and fractions.

Step 2: Meaning
We need to isolate the square root and square both sides to make the equation polynomial in terms of derivatives.

Step 3: Analysis
$y - x\frac{dy}{dx} = a\sqrt{1 + (\frac{dy}{dx})^2}$. Squaring both sides: $(y - x\frac{dy}{dx})^2 = a^2(1 + (\frac{dy}{dx})^2)$.

Step 4: Conclusion
The highest derivative is $\frac{dy}{dx}$ (order 1), and its highest power in the expanded equation is 2. Therefore, the degree is 2.
Final Answer: (D)